{
  "id": "1907.12267",
  "version": "v1",
  "published": "2019-07-29T08:23:01.000Z",
  "updated": "2019-07-29T08:23:01.000Z",
  "title": "Beta Laguerre ensembles in global regime",
  "authors": [
    "Hoang Dung Trinh",
    "Khanh Duy Trinh"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Beta Laguerre ensembles which are generalizations of Wishart ensembles and Laguerre ensembles can be realized as eigenvalues of certain random tridiagonal matrices. Analogous to the Wishart ($\\beta=1$) case and the Laguerre ($\\beta = 2$) case, for fixed $\\beta$, it is known that the empirical distribution of the eigenvalues of these ensembles converges weakly to Marchenko--Pastur distributions, almost surely. The paper restudies the limiting behavior of the empirical distribution but in regimes where the parameter $\\beta$ is allowed to vary as a function of the matrix size $N$. We show that the above Marchenko--Pastur law holds as long as $\\beta N \\to \\infty$. When $\\beta N \\to 2c \\in (0, \\infty)$, the limit is related to associated Laguerre orthogonal polynomials. Gaussian fluctuations around the limit are also studied.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-29T08:23:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "beta laguerre ensembles",
      "global regime",
      "empirical distribution",
      "associated laguerre orthogonal polynomials",
      "marchenko-pastur law holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}